Question

Two companies market new batteries targeted at owners of personal music players. DuraTunes claims a mean battery life of 11 hours, while RockReady advertises 12 hours. a) Explain why you would also like to know the standard deviations of the battery lifespans before deciding which brand to buy. b) Suppose those standard deviations are 2 hours for DuraTunes and \(1.5\) hours for RockReady. You are headed for 8 hours at the beach. Which battery is most likely to last all day? Explain. c) If your beach trip is all weekend, and you probably will have the music on for 16 hours, which battery is most likely to last? Explain.

Short answer

For 8 hours, DuraTunes is more likely to last. For 16 hours, neither is likely, but DuraTunes has a marginal edge.

Step-by-step solution

Understanding the Need for Standard Deviation

The standard deviation of the battery life is important because it indicates the variability in how long the batteries typically last. A lower standard deviation suggests that most battery lives are close to the mean, whereas a higher standard deviation indicates more variability. Knowing the standard deviation helps you understand the expected consistency and reliability of the battery performance.

Calculating Probability for 8-Hour Usage

We want to know which battery is more likely to last at least 8 hours. For DuraTunes, with a mean of 11 hours and a standard deviation of 2 hours, we calculate the z-score: \( z = \frac{8 - 11}{2} = -1.5 \). For RockReady, with a mean of 12 hours and a standard deviation of 1.5 hours, the z-score is \( z = \frac{8 - 12}{1.5} = -2.67 \). A higher z-score corresponds to a higher probability that the battery will last at least 8 hours. Hence, DuraTunes is more likely to last through the 8-hour trip.

Assessing Probability for 16-Hour Usage

For 16 hours of usage, we calculate the z-scores similarly. For DuraTunes: \( z = \frac{16 - 11}{2} = 2.5 \), and for RockReady: \( z = \frac{16 - 12}{1.5} = 2.67 \). Both z-scores correspond to a low probability, indicating that it's unlikely either battery will last this long, but DuraTunes has a slightly lower z-score, so it's marginally more likely to last, although still improbable.

Key concepts

Standard Deviation

Standard deviation is a vital concept in statistics that measures how much individual data points in a set differ from the average value, or mean, of that set. It tells us about the spread or dispersion of a set of values. In the context of the battery lifespans from DuraTunes and RockReady, the standard deviation helps us understand the reliability and predictability of the battery life performance.

Here's why it matters:

• **Low standard deviation**: Most of the values in the set are close to the mean. For batteries, this means that they are likely to perform consistently near the average advertised life.
• **High standard deviation**: Values are more spread out, indicating more variability in performance. This can mean less reliability.

Knowing the standard deviation allows consumers to predict how often a battery might not reach its average life span. Lower deviation is often preferred as it signifies more consistent performance.

Probability

Probability in statistics measures how likely an event is to occur. It is an essential tool for making decisions in uncertain situations, such as choosing which battery might last for a certain amount of time. For the battery problem, the probability is used to determine which battery is more likely to last through your day at the beach.

When looking at the batteries:

• We are calculating the likelihood of each battery lasting at least a certain amount of time.
• This is done using the z-score, which tells us how far away the desired life span (like 8 or 16 hours) is from the mean life span in terms of standard deviations.

By comparing probability, you can make an informed choice about which battery to pick for your needs. Higher probabilities indicate a greater chance of the battery lasting through your planned use.

Z-Score

The z-score is a statistic that tells us how many standard deviations a data point is from the mean. It is used to compare data points from different datasets or different situations. In our battery example, z-scores were used to calculate the probability of each battery lasting at least 8 or 16 hours.

How to compute a z-score:

• Subtract the mean from the target value (e.g. 8 or 16 hours).
• Divide by the standard deviation.

For example:- For DuraTunes with 8 hours: \(z = \frac{8 - 11}{2} = -1.5\)- For RockReady with 8 hours: \(z = \frac{8 - 12}{1.5} = -2.67\)A higher z-score indicates that the target value is less likely to deviate from the mean, suggesting a higher probability that the battery will last as needed. Z-scores help you compare and choose the best option based on the expected performance.